Non-representability of finite projective planes by convex sets

Abstract

We prove that there is no d such that all finite projective planes can be represented by convex sets in R^d, answering a question of Alon, Kalai, Matoušek, and Meshulam. As a corollary, we show that for every d there are 2-collapsible simplicial complexes that are not d-representable, strengthening a result of Matoušek and the author.

Keywords

• Convex set
• intersection pattern
• d-representable
• d-collapsible
• finite projective plane